Bragging About My Students

Two things I want to share tonight. One of them has multiple parts.


One of my international students shared a lovely gift with me yesterday. It’s a food treat that her mom sent here for her to share. I have a few food allergies so I was concerned but did not want to tell her because it felt rude. Luckily, there are a number of boys in my from who can translate the ingredient list for me. Pretty cool. Oh yeah, I’m allowed to eat it – no nuts.


I blogged a couple of days ago about a problem on my calc BC final. Here is the problem

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:


“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”


 Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty.

Twenty-four students took this final and a number of them really did a wonderful job in explaining their reasoning. I’m going to present a handful of the best responses here.

  1.  4 x 5 = 12  is number base 18. 4 x 6 = 13 is number base 21. 4 x 7 =   she will say 14 in number base 24.   4 x 12 = 19 number base 39, 4 x 13 = 20? number base 42. If following the pattern, as one of the numbers remains constant 4, another increasing by 1 each time, we get the product increasing by 1 each also. This is possible for number base 18 , increasing 3 each time. This 4 x 13 = 20 in number base 42 accordingly. However, 4 x 13 in number base 42 is one full round with another 10, which in base 42 as there exists another symbol for that suppose A, thus 4 x 13 = 1A and will never equal to 20 this way.
  2. She is using number base to calculate it. Every time 4 times initial number plus 1 and number base that is increased by 3. It will never get to twenty because the number base is always growing as number grows. Number base is growing faster than the number we multiply by. (Note – this one was accompanied by meaningful, but scribbly, calculations)
  3. (This answer starts with all the calculations hinted at in the first answer i presented)  Because the base in continually increasing by 3 and the answer is only increasing by one, the answer will never be able to get out of the ones digit.
  4. The base is increasing and in order to get to 20 the result of the calculation must be EXACTLY twice as big as the base., which is not possible.


All of these were accompanied by calculations on the side. We spent about two and a half days talking about number bases and I must admit I was really impressed by the patience that my students had with this problem. Nice way to end the year!


Thinking About How to End a Year

So, this morning both of my AP classes took their final exams. I have some questions for the world about this process, but first I want to share something fun from the Calculus BC final. You should know that my students understand that the word fun, when I use it in class, means that a problem is challenging, thought-provoking, unusual, or some other words that they might use but I won’t type. I believe that I mentioned that we ended the year after the AP test with a quick tour of some interesting topics that many high school students don’t get to see. I included a small unit on different number bases. I always start this by writing a series of addition and multiplication facts on the board. However, they don’t know that these facts are base 8 number facts. I usually reveal the secret by showing a picture of Lisa Simpson and tell them that these facts are how Lisa would compute. It’s a fun conversation to have. So, for our final exam I told my students that there would be ten problems. Nine of them would be Calculus problems taken from old tests. they have all of their old tests (all ten of them) so they could be very well prepared for that. I also told them that one problem would come from our last two weeks. After discussing this with my friend Richard – a former math teacher – he sent me the following passage from Alice in Wonderland

Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!    

I played around with this for a while and fell in love with this as a final problem for the year. This is how I presented it to my students:

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

            Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty. 


I will sink my teeth into grading finals tomorrow since I am on dorm duty tonight. I did browse through four or five of the test as they were turned in and two students really nailed the problem and provided beautiful, detailed answers explaining the pattern. I won’t spoil it here, I’ll let you work through it if you wish to do so.


I don’t know how your school works, I do know a bit about the four schools where I have worked. All of them have been independent schools that emphasize the idea that we are college preparatory schools. Each school I have worked at has had a statement in their handbook about the importance and significance of final exams as a college preparatory experience. However, I know that tonight many of the seniors in the dorm will tell me that they do not have any finals left. Today was the first of three and a half days of final exams and many seniors won’t have any more after today. There is a pretty common feeling that final exams will not be pretty and are of questionable usefulness with our seniors who are days away from graduation. This is not just a feeling at my current school. But I really wrestle with this. We say we believe that taking a final exam, preparing and organizing a large body of information for a one-day thorough examination, is a useful skill AND one that is important for college. However, it is those students who are closest to college who are the most likely to have been excused from a final exam. In some classes the final experience is a paper or a presentation that happened last week. But our school, and others where I have worked, carve out quite a bit of time for final exam administration. I wonder whether we could use our time in a more meaningful way. I wonder whether the idea of a final exam makes sense only in certain disciplines or for certain age levels. Is it reasonable for us to ask our freshmen to take exams under the same circumstances that we ask of our juniors and (sometimes) seniors? I don’t see AP scores for my students until July. I like the idea of some capstone where I check in with them in one last, broad examination of ideas. I feel pretty old-school in that regard.

I want to make my assessments meaningful for me and for my students. I am really beginning to doubt whether exam week is such a positive way to do this. I would love to hear from others about how they deal with this question. Are many of you bound to a policy that your school or your department has mandated? I want to be smarter about this and I’d love your help.

Professional Growth in a Connected Age

I’ve been teaching for a long time now. It makes me feel old when I realize I am in my 27th year in the classroom now. I joke with my students that I have been teaching longer than any of them have been alive. When I started teaching the predominant models of professional development were the inservice days at school where the school administrators decided how we needed to grow, the weekend workshops or summer workshops that I would scramble to find funding for, or the one or two day workshops that would cause me to miss school. It’s a different world now. I know I’m preaching to the choir if you are even reading this but this world of twitter, of blogs (both writing them AND reading them), of online simulcast workshops, or improv EdCamps, the list goes on. In this day in teaching I am fully convinced that if you want feedback and you want connections to help you think about your craft and to expand your toolbox – if you really want it – there is an ocean of resources at your fingertips. Literally (since I’m typing this right now!) at your fingertips. Not all of it fits everyone. I know that I am still wrestling with the timing and pace of my twitter feed, but I think I’m getting better at it and I KNOW I’m growing as a result of it. I spent a long time reading blogs, then commenting on blogs before I felt confident enough to launch my own. I have two kids at home so I know how tight time can be, but I also know that the past two Saturdays (that I blogged about separately here and here ) where I spent a combined 16 hours out of the house were worth the time and effort. Luckily Mrs. Dardy is kind and flexible and supportive of this pursuit.

I’ve been thinking quite a bit about this, about how different my life as a teacher is in the past few years. I’ve been in regular communication with one of my former colleagues, Gayle Allen. Gayle (@GAllenTC) hired me seven years ago when my family left Florida and we landed in New jersey for a while. Gayle is a remarkable, energetic thinker and was a great boss. She and I have been engaged in a long conversation about professional growth and one of the results of this conversation is an article that got posted today over at a website called Getting Smart. I know that I am not unique in this journey, but I also know that there are still many of our colleagues who have not taken this plunge. Some because they are not interested in doing so, some because they don’t know where to start. I’m pleased to be able to give shout outs of thanks to Dan Meyer (@ddmeyer) and to Sam Shah (@samjshah) through that article and I’m pleased to be connected again (even if we are nearly 3000 miles from each other) with Gayle.

This summer will see a trip to OK to take part in TwitterMathCamp. This would not have happened if Tina Cardone (@crstn85) had not reached out to me and asked me to join in on the fun. This summer will see me finish an in-house Geometry text for our students. This project would never have happened without the encouragement and advice of Jennifer Silverman (@jensilvermath). This summer will see me work on plans to help a brand new teacher in our high school take the leap from teaching Algebra I in the middle school to teaching Honors Precalculus for the first time. All of these experiences will help me grow as a professional. 27 years at it now and I feel like I still have an awful lot to learn. I hope to be smarter this time next week about this craft than I am right now.

Making Connections

We normally have very few school days between AP tests and graduation. This year we have 8 days left (counting today) so I am trying to have a little exploratory fun with my AP Calculus BC kids. They’ve worked hard on Calculus for two years now and I know that there is plenty more Calculus out there. Instead, though, I’ve chosen to take them on a little tour of some topics that they normally would not see in their high school days. Today’s topic was different number bases (the link is to my classroom document for today) and we had some fun with a series of base 8 addition and multiplication facts. I presented them with a  picture of Lisa Simpson as a clue and one of my students noticed that she only has 8 fingers. I use this as my motivation for this conversation. So, I had this nice little handout prepared that I hoped would guide us through a fun conversation. What I did not anticipate was a terrific question that came up. Let me set it up. We visited a website that converts base ten numbers to base 6 while you type. We had fun playing with it typing in things like the year of my birth and a few other nuggets. I asked them whether a base 6 representation of a number would always be longer than the base 10 representation and we had a nice chat about that. Then a student suggested that I type in a decimal so I typed 9.2 and the website did not like this. It would have been very easy to just shrug it off. My kiddos did not. They pushed me a bit and someone suggested that I write the first few negative exponents of 6. One student wisely suggested that I work on the assumption that the coefficients would all be 1 since 1/5 is SO close to 1/6. Here’s where it got fun. Another student said something along these lines – ‘Isn’t this just going to be an infinite series?’ WOW! I was so so so pleased with this. Connecting to Taylor’s and other infinite series in the face of a (relatively) harmless decimal? So proud I was.

We tried this conversion and another student seemed suspicious of the assumption that every coefficient would be 1 (or 0) so I reverted to binary. Now our task was to convert 9.2 into a binary number. The whole number part of 1001 was easy to sell. Now, we had to convert 1/5 into a series of decreasing powers of 2 (or increasingly negative powers of 2). Well, there was quite a bit of computation involved, some nice guess and check strategy was employed, the TI calculator function of turning decimals into fractions was helpful and in the end we discovered (and were able to verify) that 9.2 in base 10 is 1001.0011001100110011…

I have made a few mentions recently of my battery being recharged. It was awfully nice to see that some of my students still have some juice left in their batteries as well.

Charging My Batteries

Two Saturdays in a row now have been spent with some pretty amazing educators. Yesterday I was one of the attendees at EdCamp NEPA. One of the organizers, Mike Soskil (@msoskil) is someone I ran across on twitter some time ago. We had made plans to meet and lunch together at a tech conference but I had a minor car accident that morning so our lunch never happened. I finally met him yesterday and he was as terrific in person as he is online. He was full of energy and enthusiasm and it spread through the room. The vast majority of the folks there were first time EdCampers. I was a second timer so I felt like an old pro in this group. I’m guessing most of you reading this are familiar with the EdCamp model. You arrive to face a big blank session board and write up anything you want to talk about or listen to and people vote with their feet. Mike asked me to consider running a geogebra session and it did not take much to talk me into doing it. So I ran a session I called Visualizing Mathematics – Exploring with GeoGebra and Desmos. I showed off the work my students did in exploring the average daily temperatures in Gainesville, FL on desmos and we got into a great conversation about the power of this kind of visualization. I shared the story that all of my students missed some of the data in the same way and I got to make my standing joke about how long summer lasts in FL. One of my students asked if we would be more accurate with a city farther from the equator. Another student suggested that being closer to the equator would result in more symmetry. A conversation like this would never have happened without this visual support. I also showed off a geogebra file I created to model Taylor Series expansion as well as one of the files that the great jennifer Silverman (@jensilvermath) shared last week at our workshop. This one was called Quadratic Palooza. I had a small group in my room but they were engaged and asked me some great questions. We had a lively conversation about the power of this kind of visualization and how it enables students to ponder and ask questions that they likely would not have thought of before.

I also sat in on a couple of great sessions. The last one of the day was called Love It / Hate It. The moderator would post a statement about some school related policy/issue and we were to move to parts of the room based on whether we loved it/were on the fence / or hated it. We were to discuss with our group to construct an argument to have with our colleagues in the room.

Some of the sessions had participants taking notes together on google docs and here is the link to the schedule page


At a time of year when I am usually dragging (we have 10 class days left) I find my self with my batteries feeling recharged. Instead of feeling a bit sluggish, I am feeling perky and peppy, Love it.


A Note of Thanks

So, yesterday I had the privilege of spending about 8 hours working. I know, working on a Saturday does not necessarily sound like a privilege. But it was. I had the pleasure of working side by side with three of my colleagues who work in the same building as I do day to day. Mary and Mary and Kathy all agreed to take a Saturday away from their families and friends and join us in learning to use GeoGebra. I cannot feel them (or you) how much I appreciate their willingness to do so. I also had the pleasure of meeting three people from the Philadelphia area who all agreed to give up a Saturday AND to travel two hours to do so. Ed, and Whitney and Andy all took a chance. They heard about our workshop through a flyer that was distributed on a Philadelphia area teachers email list (thanks Ruth!) and, knowing little to nothing about me, my school, or our presenter, they chose to commit a Saturday in May to come and learn with us. I sure hope to continue hearing from them and to build some real community with them even if they are two hours away. I finally had the pleasure of meeting Justin in person! I’ve been reading his blog (and commenting there) and communicating with him on twitter. It’s been a pleasure feeling like we are building a relationship and it was a total treat to finally meet him in person. He drove from the Pittsburgh area to his mom’s house on Friday. Drove to my school and back to his mom’s house on Saturday and today he is driving back to Pittsburgh. He was full of energy, joy, and ideas yesterday and he helped make the day for me. What a treat! Lastly, I finally got to meet Jen in person. Jen has been completely generous of her time and her knowledge. I”ve been picking her brain via emails and google chats. I’ve been stealing ideas from her and she is the main reason I’m brave enough to tackle a curriculum project this summer. I’ll be writing an iBook for our Geometry course next year and I could not have conceived of doing this without her inspiration. She and her husband Charlie drove down from Connecticut and Jen was our leader in this exploration. We made some fantastic discoveries with each other (and through the help of remote twitter colleagues), we wondered and played. All of us were tired at the end and our brains hurt. That’s a great feeling, isn’t it? Community can mean many things and yesterday I felt that my community exploded beyond the walls of my school.

When My Students are More Clever than I am

This happens often enough to keep me excited in my job. I love the feeling when I learn from my students. Nearly everyday I learn important things about people, about human interactions, about kindness and community. What happened today was a great example of when I learn some math from my kiddos. I blogged yesterday about helping a boy with a L’Hopital’s Rule assignment. What I did not mention was that there was a problem I could not solve. I’m not good enough with typesetting here on wordpress so I’ll do my best here. We were trying to evaluate the right hand limit of (x – 1) * tan (pi*x/2) as x approaches 1. Two clues made me think of L’Hopital. The first is that the student told me they were working on L’Hopital’s Rule. the second was the indeterminate form of 0 *  negative infinity.

We looked at a graph to convince ourselves that there is a limit and then we tried to make this a quotient to fit the rule. My instinct with trig functions is to avoid the cotangent, cosecant, and secant functions simply because I feel more comfortable visualizing the basic cosine, since, and tangent. So I made the product a quotient by dividing the tangent expression by 1/(x-1) creating an indeterminate form of negative infinity divided by positive infinity. A quick ratio of derivatives yielded something even word and the second time around was even scarier. We walked away from the problem – in part because I was trying to do three other things at the same time and in part because he was exhausted from thinning his way through the other examples. I asked one of my best BC students to consider it overnight and had some hope that he would. 

This morning at 8 I posed this question to my quiet BC class of 7 and they pounced on it. One student advised that I rewrite the tangent function as a quotient of sine divided by cosine so that L’Hopital is immediately satisfied. Another advised that I rewrite tangent as a cotangent so that I have a ratio as well. In each case, my students saw a way to rewrite a product in terms of 0 / 0 rather than my way of infinity / infinity. Neither format is lovely but we all seemed to agree that 0 / 0 seems less scary. One round of derivatives on each idea led to the conclusion that the limit was – 2 / pi. Geogebra agreed.

What strikes me is that, despite me showing an undesirable approach AND me asking them to recall something from the past, my morning class was perfectly willing to be flexible and to TRY something. Two sound ideas in about 90 seconds and we were off. I was so delighted to start my way this day.

Why So Different?

I live in a dorm on our campus and we have about 80 boys in our building. Tonight one of them asked me to help him study for a test tomorrow in his Precalc class on conic sections. Earlier in the day at school a boy from Calculus Honors was working through L’Hopital’s Rule and struggling a bit. He was in my room working during one of my free periods and I helped him out a bit as I was running in and out of the class. In each case, I found myself really concentrating on ‘how do I do this?’ questions with these guys. Normally, I think of myself as a teacher who really focuses on process. I ask my students many questions and I try to emphasize ideas, connections, principles at work. But I find that when I am in this sort of tutoring type of environment – especially when I am working with students from other classes, not my students – then I switch to a much more functional, how do you solve this problem approach. Don’t know why I do that and I think I’m unhappy about it. I just noticed this in a very obvious way today and I want to toss that observation out to the world. Do you find yourself being a different teacher in these situations? Is there a good reason for it?